A384472 a(n) = Sum_{k=0..n} binomial(n,k)^3 * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k).

1, 2, 22, 558, 25506, 1770300, 166190354, 19647687682, 2798281247682, 466166725448544, 88942246964278060, 19127775950813311232, 4578817457796314714502, 1207681779462031251096888, 348018457509475159702959174, 108798555057988053563408904750, 36676526343321856806298038370210

Offset: 0

Author

Vaclav Kotesovec, May 30 2025

Keywords

• nonn

Comments

In general, for m > 1, Sum_{k=0..n} binomial(n,k)^m * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) ~ 2^((m+1)*n + (m-1)/2) * n^(n-(m+1)/2) / (sqrt(m-1) * Pi^((m+1)/2) * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775.

Crossrefs

Cf. A187655 (m=0), A187657 (m=1), A384471 (m=2), A384470.

Cf. A226775.

Programs

Mathematica

Table[Sum[StirlingS2[2*k, k]*StirlingS2[2*n-2*k, n-k]*Binomial[n, k]^3, {k, 0, n}], {n, 0, 20}]

Formula

a(n) ~ 2^(4*n + 1/2) * n^(n-2) / (Pi^2 * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...